I'm trying to make an optimized factotum build for a 3.P game, and I'm trying to decide on whether to take the Trivial Knowledge or Kirin's Path feat. I can not find any threads out on the Internet that compare the two feats, and I'm not terribly adapt at performing table-top math. So, could anyone give me a quick, down and dirty pro & con list for the two feats as well as their opinion on which feat is better? Thanks.

To close voters: Trivial Knowledge and Kirin's Path both do about the same thing, specifically making Knowledge checks more reliable. One involves rolling the check twice and taking the better result, the other involves taking-10 even when stressed. The question is a very simple "which one of these is mathematically better?" Definitely not too broad or opinion-based.
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KRyanJul 8 '14 at 13:42

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@KRyan Objectively 2 dice has a strictly better average result than taking-10, however whether or not removing probability from the equation is better is subjective; since any objective calculation would be based on the unknown difficulty of future rolls.
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Wesley ObenshainJul 8 '14 at 19:20

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@WesleyObenshain Yes, that is true, but that's why it's a question for Pathfinder experts, not statisticians. We know what kinds of rolls are necessary and how likely they are.
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KRyanJul 8 '14 at 19:22

@KRyan Expert opinions are still opinions is my point. They're the "good subjective" but it should still have the tag. (And many of us don't have the access level to tag it ourselves.)
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Wesley ObenshainJul 8 '14 at 19:24

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That is a misstatement. There are special considerations for sys-rec that are not automatic for other questions. Ultimately, sys-rec abides by GS/BS rules, but the community has determined that the only way to Back It Up! for sys-rec questions is with specific experience. This is not the only way to Back It Up! for other sorts of questions.
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KRyanJul 9 '14 at 2:25

3 Answers
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Mathematically, rolling twice and taking the better result is better when you want a high roll, while taking-10 is better for reliability. Basically, if taking-10 is sufficient to get the result you want, it’s the obvious choice, but if not, you need to roll to get higher. Optimizers typically treat rerolls as an effective +5, though statistically its actual value depends on your target number. Taking-10 actually lowers your average (average on a 1d20 is 10.5), but the increase in reliability is easily worth the ½.

In the case of Kirin Strike, which Kirin Path is obviously designed for, you have a pretty good chance of being able to take-10: Kirin Strike gives a +2, you should only take this feat if you have a sizeable Int score and are maximizing Knowledge ranks, and the DC is going to be 15 + CR. When facing things of CR similar to your level, 10 + ranks covers 10 + CR, more or less, so you only need to get +5 from Int and other bonuses. So taking-10 is definitely better than rolling twice for the purposes of activating Kirin Strike.

But that’s when all else is equal, which is not the case here.

Trivial Knowledge has, well, trivial requirements, just needing Int 13 (which you hopefully have if you’re focused on knowledge skills) and being a gnome (which is rarely, if ever, a bad thing; small size and +Con is awesome).

Kirin Path has brutal requirements: the skill requirements limit this to being taken at level 12 or higher, plus three feats you may or may not want (though Kirin Style isn’t terrible and Kirin Strike’s fairly solid).

Basically, you can pick up Trivial Knowledge whenever you have a free feat, while Kirin Style requires you to dedicate 40% of your feats to it. That distinction totally blows away any mathematical superiority that Kirin Path may have had. If you were getting Improved Unarmed Strike, Kirin Style, and Kirin Strike anyway it’s probably worthwhile, but I don’t really recommend those much. That’s four feats for 2×Int to damage: nice, but not that nice. Particularly since a monk is basically required to dump Int, what with their need of high Str, Dex, Con, and Wis.

What I will recommend is Knowledge Devotion from Complete Champion. It adds bonuses to attack as well as damage, the bonuses scale nicely for a variety of check results rather than being all-or-nothing, it’s a free action, and it applies to all creatures of the same kind at once. It’s a very solid feat. Plus, a dip in cloistered cleric can get it as a bonus feat by trading in their free Knowledge Domain for it. That’s alongside two other Domains or Devotions, Turn or Rebuke Undead, and a smattering of spells. A very nice choice.

You probably won't find articles or threads around the internet because Trivial Knowledge is from Races of Stone (3.5e) and Kirin Path is from Pathfinder. The major difference between the two feats is that one allows you to roll twice and take the better result, the other one allows you to simply take 10.

You can use this skill to identify monsters and their special powers or vulnerabilities. In general, the DC of such a check equals 10 + the monster's CR. For common monsters, such as goblins, the DC of this check equals 5 + the monster's CR. For particularly rare monsters, such as the tarrasque, the DC of this check equals 15 + the monster's CR, or more. A successful check allows you to remember a bit of useful information about that monster. For every 5 points by which your check result exceeds the DC, you recall another piece of useful information. Many of the Knowledge skills have specific uses as noted on Table: Knowledge Skill DCs.

My take on the two

Knowledge checks go off of your Intelligence modifier and the feats both require an Int score of 13. Which means that should you take Kirin Path your base score is 11 + the ranks you put into your knowledge skills + 2 insight + 3 if they are trained (Pathfinder skill system). You should succeed almost all of your checks by simply taking 10. The taking 10 basically negates everything except the monsters CR. The Kirin Path requires you to have unarmed strike, so you will probably have to take monk with an archtype that makes knowledge checks a class skill for you to get the most optimal usage from the feat. Trivial Knowledge does not have pre-requisits other than being a gnome and having 13 int. You do not get the insight bonuses, however you do not need to make skill rank requirements first. I think Trivial Knowledge is more flexible because you are not forced to invest so many ranks in Arcana and one of (dungeoneering, local, nature, planes, religion), however it does restrict your race.

--Trivial Knowledge--

You have the ability to dredge up obscure knowledge in appropriate situations.
Prerequisite
Gnome, Int 13,
Benefit
Whenever you make a Knowledge check or a bardic knowledge check, roll twice and use the better of the two results.

Pros: Rolling 2d20 and choosing the highest averages at a 13.82 in comparison to Kirin which is a base 10. On top of Bardic knowledge checks. http://anydice.com/

Cons: You must be a gnome, and the feat benefits Bards more so than any other class.

--Kirin Path--

Whenever you make a Knowledge check to identify a creature, even when using Kirin Style, you can take 10 even if stress and distractions would normally prevent you from doing so. While using Kirin Style against a creature you have identified using that feat, if the creature ends its turn within your threatened area, you can spend a use of your attacks of opportunity that round to move up to 5 feet times your Intelligence modifier (minimum 1). You must end your move in a square threatened by the creature. This move does not provoke attacks of opportunity.

Pros: You will almost always succeed at your knowledge checks to identify monsters. The Kirin Path feat is a chain of feats (Starting with Kirin Style then Kirin Strike).

Kirin Style - you can spend a swift action to make a Knowledge check to identify a single creature (DC 15 + creature's CR). If you succeed at the check you gain a +2 bonus on saving throws against that creature's attacks, as well as a +2 dodge bonus to AC against that creature's attacks of opportunity.

Kirin Strike - You gain a +2 insight bonus on Knowledge checks made to identify creatures, including the one Kirin Style allows. While using Kirin Style against a creature you have identified using that feat, as a swift action after you have hit a creature with a melee or ranged attack, you can add twice your Intelligence modifier in damage (minimum 2).

Cons: The feat requires: Kirin Strike, Knowledge (arcana) 12 ranks, Knowledge (dungeoneering, local, nature, planes, or religion) 5 ranks, Int 13, Kirin Style, Improved Unarmed Strike. You will most likely be playing a monk, or an archtype that has Improved Unarmed Strike, and also knowledge checks as class features.

My Opinion

I think the ultimate deciding factor comes down to your party roll. If you are playing a support / bard, then both options work. If you do plan on fighting and utilizing Unarmed Strike, then I would go with the Kirin Path. I have rolled a Bard / Monk before in pathfinder (bard - geisha/soundstriker, monk - sensai/martial artist) a character that could have used either feat. I would have chosen to go with the Trivial Knowledge (despite giving up Bardic Knowledge) because the Kirin Path is a huge feat sink, and I was investing my feats into the Improved Feint and Grapple tree.

A typical roll should be succeeded by a fully-trained PC with an average roll (10). With Kirin's Path you have an 80% chance of getting at least 10 and a 25% chance of getting at most that. Trivial Knowledge will make sure you get a 10 whenever you choose to but will make higher rolls harder to achieve, relatively speaking. So any time you need at most a 10, Trivial Knowledge is strictly better. When you don't know if 10 is enough, Kirin's Path is the safer option.